Generation of \(2\)-torsion part of Brauer group of local quintic by quaternion algebras, the totally splitting case (Q2708733)

The general problem underlying the present paper is the determination of the Brauer group \(\text{Br}(X)\) of a smooth connected projective variety \(X\) defined over a field \(k\) of characteristic \(0\). This group can be identified in a natural way with the unramified Brauer group \(\text{Br}_{ur}(K/k)\), where \(K=k(X)\) denotes the function field of \(X\) over \(k\). The setting in this paper is the following : \(k\) is a finite extension of the \(p\)-adic numbers \({\mathbb Q}_p\), and \(X\) is a smooth projective model of an affine curve \(C\) defined by an equation \(y^2=f(x)\) with \(f(x)\) a polynomial of degree \(5\) without multiple roots. The aim is to obtain a presentation of generators of \(_2\text{Br}(X)\) (the exponent-2-part of \(\text{Br}(X)\)) in terms of quaternion algebras. In the present paper, this is achieved in the case where \(f\) factors completely over \(k\). The results are very explicit but rather technical. NEWLINENEWLINENEWLINEThe case of curves of genus zero is well known, and that of elliptic curves has been treated previously by the first and the last author. These results plus further ones in the case of hyperelliptic curves which are either quintics or possess good reduction can be found in a survey paper by the authors [\textit{V. Yanchevskij}, \textit{G. Margolin} and \textit{U. Rehmann}, ``Brauer groups of curves and unramified central simple algebras over their function fields'', J. Math. Sci., New York 102, 4071-4134 (2000; Zbl 0979.14010)].NEWLINENEWLINEFor the entire collection see [Zbl 0949.00018].